Why do contractions use apostrophes when that means they aren’t any shorter (replacing a letter with an apostrophe makes it still the same number of characters)?

That is not a correct assumption. For example, the word cannot, when contracted, becomes “can’t,” saving one character. The words “do not“ become “don’t," saving one character and one space. The apostrophe character is narrower than the character or space it replaces. Contractions become especially useful when speaking, because they usually save one or more syllables in each word or group of words.